The study of material fatigue analysis is crucial as durable goods last longer. Early fatigue analysis began in the 1920s and 30s with aircraft engines. Technicians mounted engines on towers and ran them until failure occurred. Initial designs were conservative, producing about ½ hp per cubic inch. As time passed, designs improved, achieving over 1 hp per cubic inch in a reliable manner.
Materials vary in their fatigue properties. For example, aluminum can fail due to fatigue, even under light loads. Documentaries show intact planes sent to scrap yards. There, they remove useful parts and destroy the rest. These aluminum structures face constant stress from takeoffs, flights, and landings. This limits how many cycles they can handle.
In the 1950s, the deHavilland Comet, a British jet airliner, faced a major fatigue failure. Several Comets broke apart mid-flight due to cracks from rivet holes in the fuselage. The holes were punched, causing fine cracks. Each pressurization cycle and turbulence stressed the holes until they burst. Steel exhibits varying behavior depending on its alloy composition. General fatigue stress rules suggest using 1/4 to 3/4 of the yield stress to find the infinite life limit.
A recent failure occurred in Minneapolis on a bridge over the Mississippi River. Built in the late 1960s, the bridge’s load increased with more lanes. Ordinary steel alloys are common. Fatigue stress is usually about one-fourth or less of the yield stress. The gusset plates connecting support beams failed with a sudden and tragic collapse. Fatigue failures often give little warning. Chemical corrosion from salt exposure also affects bridges, damaging both steel and concrete.
Fatigue Analysis Methods
Here are examples of High Cycle Fatigue Analysis methods.
Miner’s Rule
Miner’s rule is a simple cumulative damage model. If there are k different stress levels, and the average cycles to failure at the ith stress, Si, is Ni, then the damage fraction, C, is:
C = Σ(ni/Ni)
where:
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ni is the number of cycles at stress Si.
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C is the fraction of life consumed at different stress levels.
When C reaches 1, failure occurs. This equation determines how much life each stress level uses. It adds the parts together. Often, we can quantify damage as the product of stress and cycles under that stress.
C = Σ(ni/ Wi)
Assuming critical damage is the same across all stress levels:
For instance, if WFailure = 50 for a component, it will fail after 10 cycles at stress level 5 or 25 cycles at stress level 2. Using the critical value of damage, Eqn. (1) becomes:
C shows the cumulative damage proportion to the critical value.
Example:
If a part spends 10% of its life at alternating stress level σ1, 30% at σ2, and 60% at σ3, how many cycles, n, can it undergo before failure? From the S-N diagram, if we know Ni for each stress, we observe that failure occurs when:
C = (n1/N1) + (n2/N2) + (n3/N3) = 1
Note:
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For “high-low” fatigue tests, where testing occurs sequentially at two stress levels (σ1, σ2), failure happens when C is usually < 1. For “low-high” tests, ***C*** values tend to be > 1.
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Random loading histories correlate well with the Palmgren-Miner rule.
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The Palmgren-Miner rule can graphically shift the S-N curve. If N1 cycles are applied at stress level σ1, the S-N curve shifts to a new life value, N’1.
Limitations
The Palmgren-Miner rule has a key limitation: it ignores sequence effects. This means the order of loading doesn’t matter. Many cases observe sequence effects. Another limitation is that it treats damage accumulation as independent of stress level. The modified S-N diagram shows that the entire curve shifts in a uniform manner, regardless of the stress amplitude.
Goodman Relation
The Goodman relation in materials science and fatigue analysis shows how mean and alternating stresses affect a material’s fatigue life.
A Goodman diagram, or Haigh diagram, shows mean stress versus alternating stress. It highlights failure points across cycles. Experimental data on these plots often creates a parabola called the Gerber line. This can be roughly estimated using a straight line known as the Goodman line.
Mathematically, the Goodman relation can be represented as:
σa / σf’ + σm / σut = 1
where:
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σa is the alternating stress,
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σm is the mean stress,
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σf’ is the fatigue limit for completely reversed loading,
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σut is the ultimate tensile stress of the material.
As mean stress goes up, fatigue life goes down for a given applied stress. Plotting this relation shows safe cyclic loading. If the mean and applied stress points are below the curve, the part will last. If they are above, it will fail under those stress conditions.
Low Cycle Fatigue – Coffin-Manson Relation
High stress that leads to plastic deformation makes using stress for loading assessment less effective. The material’s strain offers a simpler, more accurate description. Low-cycle fatigue is often characterized by the Coffin-Manson relation:
Δεp/2 = εf’ (2N)^c
where:
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Δεp/2 is the plastic strain amplitude,
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εf’ is the fatigue ductility coefficient (the failure strain for one reversal),
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2N is the number of reversals to failure (N cycles),
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c is the fatigue ductility exponent, usually ranging from -0.5 to -0.7 for metals in time-independent fatigue. In cases with creep or environmental interactions, slopes can be steeper.
Norman T. Neher, P.E.
Analytical Engineering Services, Inc.
Elko New Market, MN
www.aesmn.org